Automatically assign product category based on assigned product tag

I want to automatically assign product category based on assigned product tag. I want to do exactly what this ( https://wordpress.org/plugins/auto-assign-post-category/ ) plugin does, but with products, not posts. I have also this code:

function auto_add_category ($product_id = 0) {

    if (!$product_id) return;

    // because we use save_post action, let's check post type here
    $post_type = get_post_type($post_id);
    if ( "product" != $post_type ) return;

    $tag_categories = array (
        'Biblioteka' => 'Biblioteki',
        'Biurko' => 'Biurka',
        'Fotel' => 'Fotele',
        'Komoda' => 'Komody',
        'Konsola' => 'Konsole',
        'Kredens' => 'Kredensy',
        'Krzesło' => 'Krzesła',
        'Lustro' => 'Lustra',
        'Ława' => 'Ławy',
        'Łóżko' => 'Łóżka',
        'Regał na książki' => 'Regały na książki',
        'Sofa' => 'Sofy',
        'Stolik' => 'Stoliki',
        'Stół' => 'Stoły',
        'Stół ogrodowy' => 'Stoły ogrodowe',
        'Szafka nocna' => 'Szafki nocne',
        'Szafka RTV' => 'Szafki RTV',
        'Szafa' => 'Szafy',
        'Toaletka' => 'Toaletki',
        'Witryna' => 'Witryny',
    );

    // get_terms returns ALL terms, so we have to add object_ids param to get terms to a specific product
    $product_tags = get_terms( array( 'taxonomy' => 'product_tag', 'object_ids' => $product_id ) );
    foreach ($product_tags as $term) {
        if ($tag_categories[$term->slug] ) {
            $cat = get_term_by( 'name', $tag_categories[$term->slug], 'product_cat' );
            $cat_id = $cat->term_id;
            if ($cat_id) {
                $result =  wp_set_post_terms( $product_id, $cat_id, 'product_cat', true );
            }
        }
    }
}
add_action('publish_post','auto_add_category');

But I can't make it work. Products are already on site. Thanks for help!